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# The Cross-Section of Volatility and Expected Returns

BlackRock

^{1}, Columbia University^{2}, National Bureau of Economic Research^{3}, Rice University^{4}, Tsinghua University^{5}TL;DR: This paper examined the pricing of aggregate volatility risk in the cross-section of stock returns and found that stocks with high sensitivities to innovations in aggregate volatility have low average returns, and that stock with high idiosyncratic volatility relative to the Fama and French (1993) model have abysmally low return.

Abstract: We examine the pricing of aggregate volatility risk in the cross-section of stock returns Consistent with theory, we find that stocks with high sensitivities to innovations in aggregate volatility have low average returns In addition, we find that stocks with high idiosyncratic volatility relative to the Fama and French (1993) model have abysmally low average returns This phenomenon cannot be explained by exposure to aggregate volatility risk Size, book-to-market, momentum, and liquidity effects cannot account for either the low average returns earned by stocks with high exposure to systematic volatility risk or for the low average returns of stocks with high idiosyncratic volatility

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TL;DR: It is found that EPU positively forecasts log excess market returns and innovations in EPU earn a significant negative risk premium in the Fama-French 25 size-momentum portfolios.

Abstract: Using the Baker, Bloom, and Davis (2013) news-based measure to capture economic policy uncertainty (EPU) in the United States, we find that EPU positively forecasts log excess market returns. A one-standard deviation increase in EPU is associated with a 1.5% increase in forecasted 3-month abnormal returns (6.1% annualized). Furthermore, innovations in EPU earn a significant negative risk premium in the Fama French 25 size-momentum portfolios. Among the Fama French 25 portfolios formed on size and momentum returns, the portfolio with the greatest EPU beta underperforms the portfolio with the lowest EPU beta by 5.53% per annum, controlling for exposure to the Carhart four factors as well as implied and realized volatility. These findings suggest that EPU is an economically important risk factor for equities.

800 citations

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TL;DR: In this paper, a five-factor model that adds profitability (RMW) and investment (CMA) factors to the three factor model of Fama and French (1993) suggests a shared story for several average-return anomalies.

Abstract: A five-factor model that adds profitability (RMW) and investment (CMA) factors to the three-factor model of Fama and French (1993) suggests a shared story for several average-return anomalies. Specifically, positive exposures to RMW and CMA (returns that behave like those of the stocks of profitable firms that invest conservatively) capture the high average returns associated with low market β, share repurchases, and low stock return volatility. Conversely, negative RMW and CMA slopes (like those of relatively unprofitable firms that invest aggressively) help explain the low average stock returns associated with high β, large share issues, and highly volatile returns.

605 citations

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TL;DR: This paper used a sample of option prices and the method of Bakshi, Kapadia and Madan (2003) to estimate the ex ante higher moments of the underlying individual securities' risk-neutral returns distribution.

Abstract: We use a sample of option prices, and the method of Bakshi, Kapadia and Madan (2003), to estimate the ex ante higher moments of the underlying individual securities’ risk-neutral returns distribution. We find that individual securities’ volatility, skewness, and kurtosis are strongly related to subsequent returns. Specifically, we find a negative relation between volatility and returns in thecross-section. We also find a significant relation between skewness and returns, with more negatively (positively) skewed returns associated with subsequent higher (lower) returns, while kurtosis is positively related to subsequent returns. We analyze the extent to which these returns relations represent compensation for risk. We find evidence that, even after controlling for differences in comoments, individual securities’ skewness matters. As an application, we examine whether idiosyncratic skewness in technology stocks might explain bubble pricing in Internet stocks. However, when we combine information in the risk-neutral distribution and a stochastic discount factor to estimate the implied physical distribution of industry returns, we find little evidence that the distribution of technology stocks was positively skewed during the bubble period – in fact, these stocks have the lowest skew, and the highest estimated Sharpe ratio, of all stocks in our sample.

594 citations

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TL;DR: The authors investigated the relation between global foreign exchange (FX) volatility risk and the cross-section of excess returns arising from popular strategies that borrow in low-interest rate currencies and invest in high interest rate currencies, so-called carry trades.

Abstract: We investigate the relation between global foreign exchange (FX) volatility risk and the cross-section of excess returns arising from popular strategies that borrow in low-interest rate currencies and invest in high-interest rate currencies, so-called 'carry trades'. We find that high interest rate currencies are negatively related to innovations in global FX volatility and thus deliver low returns in times of unexpected high volatility, when low interest rate currencies provide a hedge by yielding positive returns. Our proxy for global FX volatility risk captures more than 90% of the cross-sectional excess returns in five carry trade portfolios. In turn, these results provide evidence that there is an economically meaningful risk-return relation in the FX market. Further analysis shows that liquidity risk also matters for expected FX returns, but to a lesser degree than volatility risk. Finally, exposure to our volatility risk proxy also performs well for pricing returns of other cross sections in foreign exchange, U.S. equity, and corporate bond markets.

580 citations

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TL;DR: In this paper, the predictive relationship between social media and firm equity value, the relative effects of social media metrics compared with conventional online behavioral metrics, and the dynamics of these relationships were examined.

Abstract: Companies have increasingly advocated social media technologies to transform businesses and improve organizational performance. This study scrutinizes the predictive relationships between social media and firm equity value, the relative effects of social media metrics compared with conventional online behavioral metrics, and the dynamics of these relationships. The results derived from vector autoregressive models suggest that social media-based metrics (web blogs and consumer ratings) are significant leading indicators of firm equity value. Interestingly, conventional online behavioral metrics (Google searches and web traffic) are found to have a significant yet substantially weaker predictive relationship with firm equity value than social media metrics. We also find that social media has a faster predictive value, i.e., shorter “wear-in” time, than conventional online media. These findings are robust to a consistent set of volume-based measures (total blog posts, rating volume, total page views, and search intensity). Collectively, this study proffers new insights for senior executives with respect to firm equity valuations and the transformative power of social media.

475 citations

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TL;DR: In this paper, the authors present some additional tests of the mean-variance formulation of the asset pricing model, which avoid some of the problems of earlier studies and provide additional insights into the nature of the structure of security returns.

Abstract: Considerable attention has recently been given to general equilibrium models of the pricing of capital assets Of these, perhaps the best known is the mean-variance formulation originally developed by Sharpe (1964) and Treynor (1961), and extended and clarified by Lintner (1965a; 1965b), Mossin (1966), Fama (1968a; 1968b), and Long (1972) In addition Treynor (1965), Sharpe (1966), and Jensen (1968; 1969) have developed portfolio evaluation models which are either based on this asset pricing model or bear a close relation to it In the development of the asset pricing model it is assumed that (1) all investors are single period risk-averse utility of terminal wealth maximizers and can choose among portfolios solely on the basis of mean and variance, (2) there are no taxes or transactions costs, (3) all investors have homogeneous views regarding the parameters of the joint probability distribution of all security returns, and (4) all investors can borrow and lend at a given riskless rate of interest The main result of the model is a statement of the relation between the expected risk premiums on individual assets and their "systematic risk" Our main purpose is to present some additional tests of this asset pricing model which avoid some of the problems of earlier studies and which, we believe, provide additional insights into the nature of the structure of security returns The evidence presented in Section II indicates the expected excess return on an asset is not strictly proportional to its B, and we believe that this evidence, coupled with that given in Section IV, is sufficiently strong to warrant rejection of the traditional form of the model given by (1) We then show in Section III how the cross-sectional tests are subject to measurement error bias, provide a solution to this problem through grouping procedures, and show how cross-sectional methods are relevant to testing the expanded two-factor form of the model We show in Section IV that the mean of the beta factor has had a positive trend over the period 1931-65 and was on the order of 10 to 13% per month in the two sample intervals we examined in the period 1948-65 This seems to have been significantly different from the average risk-free rate and indeed is roughly the same size as the average market return of 13 and 12% per month over the two sample intervals in this period This evidence seems to be sufficiently strong enough to warrant rejection of the traditional form of the model given by (1) In addition, the standard deviation of the beta factor over these two sample intervals was 20 and 22% per month, as compared with the standard deviation of the market factor of 36 and 38% per month Thus the beta factor seems to be an important determinant of security returns

2,899 citations

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TL;DR: In this paper, the generalized autoregressive conditionally heteroskedastic (GARCH) model of returns is modified to allow for volatility feedback effect, which amplifies large negative stock returns and dampens large positive returns, making stock returns negatively skewed and increasing the potential for large crashes.

Abstract: It is sometimes argued that an increase in stock market volatility raises required stock returns, and thus lowers stock prices. This paper modifies the generalized autoregressive conditionally heteroskedastic (GARCH) model of returns to allow for this volatility feedback effect. The resulting model is asymmetric, because volatility feedback amplifies large negative stock returns and dampens large positive returns, making stock returns negatively skewed and increasing the potential for large crashes. The model also implies that volatility feedback is more important when volatility is high. In U.S. monthly and daily data in the period 1926-88, the asymmetric model fits the data better than the standard GARCH model, accounting for almost half the skewness and excess kurtosis of standard monthly GARCH residuals. Estimated volatility discounts on the stock market range from 1% in normal times to 13% after the stock market crash of October 1987 and 25% in the early 1930's. However volatility feedback has little effect on the unconditional variance of stock returns.

1,793 citations

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TL;DR: In this article, the authors examined a class of continuous-time models that incorporate jumps in returns and volatility, in addition to diffusive stochastic volatility, and developed a likelihood-based estimation strategy and provided estimates of model parameters, spot volatility, jump times and jump sizes using both S&P 500 and Nasdaq 100 index returns.

Abstract: This paper examines a class of continuous-time models that incorporate jumps in returns and volatility, in addition to diffusive stochastic volatility. We develop a likelihood-based estimation strategy and provide estimates of model parameters, spot volatility, jump times and jump sizes using both S&P 500 and Nasdaq 100 index returns. Estimates of jumps times, jump sizes and volatility are particularly useful for disentangling the dynamic effects of these factors during periods of market stress, such as those in 1987, 1997 and 1998. Using both formal and informal diagnostics, we find strong evidence for jumps in volatility, even after accounting for jumps in returns. We use implied volatility curves computed from option prices to judge the economic differences between the models. Finally, we evaluate the impact of estimation risk on option prices and find that the uncertainty in estimating the parameters and the spot volatility has important, though very different, effects on option prices.

1,040 citations

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TL;DR: In this article, a new way to generalize the insights of static asset pricing theory to a multi-period setting is proposed, which uses a loglinear approximation to the budget constraint to substitute out consumption from a standard intertemporal asset pricing model.

Abstract: This paper proposes a new way to generalize the insights of static asset pricing theory to a multi-period setting. The paper uses a loglinear approximation to the budget constraint to substitute out consumption from a standard intertemporal asset pricing model. In a homoskedastic lognormal selling, the consumption-wealth ratio is shown to depend on the elasticity of intertemporal substitution in consumption, while asset risk premia are determined by the coefficient of relative risk aversion. Risk premia are related to the covariances of asset returns with the market return and with news about the discounted value of all future market returns.

805 citations

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TL;DR: This article investigated whether market-wide liquidity is a state variable important for asset pricing and found that expected stock returns are related cross-sectionally to the sensitivities of returns to fluctuations in aggregate liquidity.

Abstract: This study investigates whether market-wide liquidity is a state variable important for asset pricing. We find that expected stock returns are related cross-sectionally to the sensitivities of returns to fluctuations in aggregate liquidity. Our monthly liquidity measure, an average of individual-stock measures estimated with daily data, relies on the principle that order flow induces greater return reversals when liquidity is lower. Over a 34-year period, the average return on stocks with high sensitivities to liquidity exceeds that for stocks with low sensitivities by 7.5% annually, adjusted for exposures to the market return as well as size, value, and momentum factors.

789 citations